X. PHYSICAL ACOUSTICS

Academic Research Staff

Prof. K. U. Ingard

Graduate Students

A. A. Moretta V. K. SinghalS. R. Rogers J. A. Tarvin

RESEARCH OBJECTIVES

Our general objective involves the study of the emission, propagation, and absorp-tion of acoustic waves in matter. Specific areas of present research include (i) theinteraction of waves with coherent light beams in fluids and solids, (ii) nonlinear acous-tics in fluids, and (iii) the generation and propagation of sound waves in moving fluids,with particular emphasis on waves propagated in ducts and generation of sound byturbulent flow.

K. U. Ingard

A. UPSTREAM AND DOWNSTREAM SOUND RADIATION INTO

A MOVING FLUID

ONR (Contract N00014-67-A-0204-0019)

K. U. Ingard, V. K. Singhal

The problem of sound radiation from a source in relative motion with respect to the

surrounding fluid has become of considerable interest, particularly in connection with

noise generation in aircraft and in various types of fluid machinery. Although the basic

effects, as well as many details of the influence of fluid motion on sound radiation, have1-4been identified by several investigators, these studies have been limited to mathemat-

ical analysis of the idealized case of moving point sources. The problem of sound

emission from real sources of finite dimensions has received much less attention. We

present an explicit demonstration of the influence of relative fluid motion on sound radi-

ation from a stationary source, with particular attention to the relation between the

sound pressure fields radiated upstream and downstream.

1. Experimental Arrangement

The experimental arrangement is shown schematically in Fig. X-1. A sound source

is mounted in one of the side walls of a rectangular duct with inner dimensions

This work is supported by the U. S. Navy - Office of Naval Research (ContractN00014-67-A-0204-0019) and by the Joint Services Electronics Programs (U. S. Army,U.S. Navy, U.S. Air Force) under Contract DAAB07 -71-C -0300).

QPR No. 108

(X. PHYSICAL ACOUSTICS)

3/4 X 7/8 in. The duct is connected to a steam ejector through a plenum chamber, and

the flow speed in the tube can be varied from 0 to -100 m/s, which corresponds to Mach

number ~0. 3.

The loudspeaker is driven by means of a pulse generator and produces harmonic

sound-pressure wave trains in the duct. The carrier frequency of these waves is chosen

to be considerably lower than the cutoff frequency for the first higher order acoustic

mode in the duct, so that at the carrier frequency only the plane-wave mode will be able

to propagate.

The pressure pulses are detected by two identical pressure transducers mounted in

the side walls of the duct on the upstream and downstream sides of the sound source at

equal distances from it. In the absence of flow, M = 0, the recorded pulses from these

transducers are simultaneous and identical in shape, as demonstrated by the results

shown in Fig. X-1. Any difference in the sensitivity of the transducers is compensated

for by gain adjustment of the transducer amplifiers, so that in the absence of flow the

amplitudes of the recorded pulses from the transducers are equal.

P_M=0.45

Fig. X-1.

-- FLOW / Recorded pressure pulses in the upstream (p_)TRANSDUCER TRANSDUCER and the downstream (p ) directions at flow Mach

SOUND SOURCE +

numbers I = 0 and M = 0. 3.P

P,/ M=0

When the air in the duct is moving, however, the pressure amplitudes are

no longer equal; the amplitude in the upstream direction is larger than in the

downstream direction, and the difference increases monotonically with the flow

speed in the duct. An example of recorded upstream and downstream pressure

waves at a flow Mach number 0. 3 is shown in Fig. X-1. In addition to the

obvious difference in the pressure amplitudes, a difference in the time of arrival

of the two pulses is also apparent in this figure. From this time difference

and the known distance between the sound source and the receivers, the flow

Mach number in the duct can be determined.

The Mach number dependence of the ratio between the upstream and downstream

pressure amplitudes obtained in this manner is illustrated in Fig. X-2. Measurements

were carried out at two frequencies, 800 Hz and 2000 Hz. As can be seen, there is

no marked difference in the amplitude ratio at these frequencies.

QPR{ No. 108

(X. PHYSICAL ACOUSTICS)

* /

S (I+M)

3.0

P2

2.5

2.0

/++0+

0 0.1

+-+++ +'t"

+ 1000 Hz* 1400 Hz

0.4 0.5

Fig. X-2. \Mach number dependence of the measured ratio P_/P+ between the pres-

sure amplitudes radiated in the upstream and downstream directions.

2. Mathematical Analysis

In the mathematical analysis of this problem we start from the wave equation for

the sound pressure field p(x, y, z, t)

2 2 2 2 22 a 2p a2p a P 2M a2p a2 p

(1-M ) + + 2 2 0, (1)ax2 ay2 az2 c axatc2 a12Dx Dy Dz c at

where c is the sound speed, and M the Mach number. The coordinates x, y, z refer to

a stationary laboratory frame of reference with respect to which the unperturbed fluid

is assumed to move with uniform speed Mc in the positive x direction. We wish to

solve this equation, subject to the boundary conditions peculiar to our experimental

arrangement.

The duct walls, placed in the planes y = 0, y = a and z = 0, z = b, are assumed to

be rigid everywhere except in the source region, as indicated in Fig. X-3. Conse-

quently, the normal components of the fluid velocity and the corresponding pressure

gradients are zero at the boundaries except at the.source. If the source, located in the

wall in the plane y = 0, produces a velocity perturbation u in the fluid flow in the plane

y = 0, the effect of the source can be expressed as the boundary condition

u = u f(x, z, t)y 0 (y= 0).

QPR No. 108

0 //

//(I+M )2

(X. PHYSICAL ACOUSTICS)

SOURCE REGIONuy= uy (x, z, t)

Fig. X-3. The source region is in the plane y = 0 of the duct and is definedby the perturbation of the fluid velocity u in the duct.

It is important to realize, however, that this velocity perturbation of the fluid in the

duct is not necessarily the same as the velocity of the oscillating air column in the

throat of the loudspeaker source

ar s(x, z, t)us t (3)

5 at

where Trs (x, z, t) is the displacement of the air column. Although u = us is valid when

there is no mean flow in the duct, the situation is more complex when mean flow is

present. For example, if the flow over the source region is streamlined, the trans-

verse oscillatory motion of the air out of the source will result in a displacement of

the streamlines, and this displacement gives rise to a velocity perturbation in the fluid

flow given by

u (x, z, t) = + Mc s(x, z, t). (4)

This model of the flow perturbation produced by the source may not be quite realistic

in a highly turbulent duct flow, in which case the contribution from the space derivative

in Eq. 4 is expected to be reduced by the irregularities in the flow. In the absence of

this contribution, the boundary condition in (4) reduces to u = u .y sIn our experiment, since only the plane-wave mode is transmitted along the duct,

it is expedient to introduce the average sound pressure p over the duct cross section,

p = f p dydz. (5)

We now obtain a wave equation for p by integrating Eq. 1 over the transverse coordi-

nates y, z. We make use of the fact that the duct walls are rigid, except in the

QPR No. 108

(X. PHYSICAL ACOUSTICS)

2 2source region, and note that the average of p/a z is zero. Similarly, the average of

a2p/ay 2 is (1/A) f (-Dp/Dy) 0 dz, where (ap/ay) 0 is evaluated in the source region of the

wall at y = 0. We can express (ap/ay) 0 in terms of the velocity perturbation u from

the momentum equation

P t + M fx Uy ay"

The wave equation for the average pressure p can then be expressed as

2- 2- 2-p Nap a p

(1-M - - = s(x, t),ax2 c axatc2 at2 2ax c at

where

P t) = - ) b y(x, ,t)dz.S(X, t) + a A C a UY (x, z, t) d z.at ax o

In this inhomogeneous wave equation the right-hand side is considered to be a known

source function s(x, t) defined in the source plane y = 0, where u is given by Eq. 4. ToY

solve this equation, we introduce the Fourier transforms

ikx -iwtp(x, t) = ff P(k, w) e e dkdw

ikx -iwts(x, t) = ff S(k, w) e e dkdw,

and from Eq. 7 obtain

-S(k, w)P(k, e) = -M 2w(erk)(k (+ _

where

k = W+ c(kc(1+M)

(10)

(11)

(12)k - wMc(l-M)

If we let the source region be limited to -L < x < L, so that s(x, t) is zero outside

this region, we can express S(k, o) as

S(k, w) = I +7T -L

s(x , co) exp(-ikx ) dx ,0 0

QPR No. 108

(13)

(8)

(X. PHYSICAL ACOUSTICS)

where s(x, w) is the temporal Fourier transform of s(x, t). Then, from Eqs. 4 and 7,

we have

S(k, o) = (iw) ()pu(1 - e (k , (14)

where

b L

is (k, 1) z 0 sL zs(x,' °, co ) exp(-ikx°) dxo (15)o -L

and A = b2L is the source area, uo = (-iuo) is the velocity amplitude of the source,

and ro = the displacement amplitude. In the special case of a pistonlike displacement

such that rs (xo z, C) = no in the source region, we have

sin kL (16)ps(k, w) kL (16)

Having obtained S(k, o), we obtain the pressure amplitude p(x, w) from

ikx

p(x, ) P(k, co) eikx dk = (k, dk, (17)-o00 - e (1-MA ) (k - k + )(k - k )

where S(k, c) is given by Eqs. 14 and 15.

The poles k = k and k = k , although located on the k axis in the present analysis,

would contain a small positive and negative part, respectively, if some damping mech-

anism, such as viscosity or heat conduction, had been included in the analysis. Eval-

uating the integral by contour integration in the complex k plane, we can complete the

contour in the upper k plane for x > L, and thus include the pole at k = k . The corre-

sponding solution is then

p= A (pcu 2 s(k+, ~ ) exp x . (18)+0 (1+ M) c(1+ M)

Similarly, by closing the path in the lower half plane, we find, for x < -L,

s 1 1 .(o

P A (pcuo 2 s(k c ) exp 1 x . (19)(A 21-) c(1-M7)

These solutions represent the waves transmitted in the downstream and upstream direc-

tions traveling with speeds c(1+M) and c(l-M), respectively, as expected. It is inter-

esting to note that the amplitudes of these waves are different in the presence of flow in

QPR No. 108

(X. PHYSICAL ACOUSTICS)

the duct. If the source region is acoustically compact, that is, if L is much smaller

than the acoustic wavelength X, the value of the source function s is close to unity, as

can be seen in the special example given in Eq. 16, and the ratio of the upstream and

downstream pressure amplitudes becomes

p_ (l+M) 2

(20)p (1-M )

2 "

3. Discussion

It should be emphasized that the Mach number dependence of the wave amplitudes,

expressed by the factors (1+M)-2 and (I-M)-2 in Eqs. 18 and 19 and leading to the

amplitude ratio in Eq. 20, depends intimately on the nature of the source and the velocity

perturbation that it produces in the fluid. In the analysis carried out here the relation-

ship between the (known) displacement of the air column in the loudspeaker throat and

the corresponding velocity perturbation produced in the duct flow has been assumed to

be described by Eq. 4, a relation based on the model of an oscillatory displacement of

streamlined flow over the source region. In highly turbulent flow it may be more real-

istic to use as a boundary condition u = u = as /at (obtained by neglecting a/ax iny s

Eq. 4), which means that the velocity perturbation in the duct flow equals the velocity

in the loudspeaker throat. The amplitude ratio p /p obtained in this case is

In Fig. X-2, which shows the measured amplitude ratio Ip_ /p+ as a function of the

flow Mach number, we have also plotted the functions F 1 (M) = (1+M)2/(I-M)2 and

F 2 (M) = (1+M)/(1-M), which represent the theoretical results obtained on the basis of

the two different boundary conditions that were considered. It is interesting to find that

the experimental results fall between these theoretical curves. At Mach numbers less

than -~0. 1, the data are in good agreement with the function F 2, which indicates that

the streamline model of the flow is meaningful. As the Mach number is increased, how-

ever, the data show a trend toward the function F 1 , which favors the boundary condition

U =u.y s

The experiments were carried out at the M. I. T. Gas Turbine Laboratory, and we

wish to thank Angelo Moretta for assistance in setting up the flow facility for the exper-

iments.

References

1. H. Billing, Trans. Max Planck Institute fiir Stromungsforschung, No. 26, p. 231,1948.

2. D. I. Blokhintsev, "Acoustics of a Nonhomogeneous Moving Medium," NACA TM1399, February 1956 (translation).

QPR No. 108

(X. PHYSICAL ACOUSTICS)

3. M. J. Lighthill, Proc. Roy. Soc. (London) A 222, 1-32 (1954).

4. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill Book Company,New York, 1968), pp. 682-689.

B. SOUND ABSORPTION BY A SINGLE RESONATOR IN A DUCT

ONR (Contract N00014-67-A-0204-0019)

A. G. Galaitsis

1. Introduction

The absorption characteristics of an acoustic resonator as a duct termination have

been studied in detail.1, 2 Here we consider the influence of a nonlinearly responding

resonator attached to a side wall of a duct on a sound wave propagating along the duct.

The basic features of the problem are depicted in Fig. X-4. A plane sound

wave pi propagates to the left along the duct D and excites the resonator C which com-

municates with the duct through the orifice O. Consequently, a fraction of the incident

wave is dissipated in the resonator, another part, pr, is reflected back toward the sound

source and the rest, pt, propagates freely beyond the resonator.

We shall first derive expressions for the transmitted and reflected fractions of the

incident energy and compare them with measured values.

2. Theory

Consider the system in Fig. X-4. The radius and cross section area of the orifice

are r0 and A , the width and cross section of the duct are d and A, and the wavelength

of the sound wave is X. We shall assume that d, ro << , so that we deal only with plane

C

O/o D

+ o

UL uR Pt

Pr

Fig. X-4. Acoustic resonator connected in parallel to a duct.

waves. In the absence of reflection at the far end of the duct the total fields pL and PH

to the left and right of the resonator are

QPh No. 108

(X. PHYSICAL ACOUSTICS)

pL o(eikx + R e-ikx) -iwtPL po(e +l e ) e

ikx-ictP = poT e

and the corresponding acoustic velocities are

Po (eikx -ikx -iwtu c(e -Re )eL pc

Po ikx-iwtUR =-Te

R pc

The coefficients R and T are obtained by combining the boundary conditions

u - [UL-UR

(x= 0)

PL = PR

with the equation giving the response of the resonator

PR = pcSoUo (x = 0),

where u is the acoustic velocity at the orifice, a = Ao/A, and 0 (the impedance of the

orifice and resonator combination) is given by

uo= e - ix = 11 - + eo

0 c o - i(X + ,).o r

Expression (3) contains the linear contributions 0o' X0,

linear contribution proportional to

U1 o'"

We substitute (1) and (2) in (3)-(5), and get

2o + o

3and Xr, as well as the non-

20T-2{ + O

Here R and T depend on u 1 through o .Therefore ul must be calculated before (6) can

be evaluated. We calculate u l by observing that

SpR (x= 0)

pc o01

QPR No. 108

-L

SPL too;d130 7N".

t40

-- K

FREQUENCY (Hz)

1.0

4,O

S A

TRANSMtTTED

. , .- ..

S1

FREQUENCY (Hz)

Fig. X-5. Predicted reflected (RE) and transmitted (TR) fractions of theincident power.

QPR No. 108

till

___ __

I .2

T AN MITTED.- : , TTr:

FREQUENCY (Hz)

Fig. X-6. Measured reflected (RE) andincident power.

transmitted (TR) fractions of the

QPR No. 108

9 80 0FREQUENCY (Hz)

L.L"L 1, 14 1

:-":~ c_, - ,. =-=i=,

i; < : < TU< :r::.2: :: : : :--:1.i, : :T:.,4.

1 10'

1 - i

Im

-1 I: 1

k

i.

(X. PHYSICAL ACOUSTICS)

which leads to

24 3 2 22 2q 1 l + 2ql ou + (0 + X )u 1 2 = 0,

with q1 = 1. 1/c and q 2 = p 2 /pc and p, = pR(x= 0). The roots of (8) are obtained by com-

puter and R and T are subsequently evaluated. The results are shown in Fig. X-5,

where RE = R and TR = Ti for different values of p 2 .

3. Experiment

The experimental arrangement for measuring RE and TR has been described in a

previous report. 4 The experimental values for TR and RE were obtained by measuring

the maximum and minimum values of the standing waves to the left (H x , Hn) and right

(h x , hn) of the acoustic resonator and then taking

h +hnTR = h x n

H +H

2RE= Hx + Hn

x n

for different values of p2 . The results are shown in Fig. X-6. The agreement is sat-

isfactory and the nonlinear nature of the interaction is obvious in both cases. As the

amplitude of the driving field increases, the impedance of the resonator increases

because of the presence of the 1. 1 ul/c term, and the absorption decreases. Conse-

quently a greater fraction of the power is transmitted.

References

1. U. Ingard and H. Ising, J. Acoust. Soc. Am. 42, 6-17 (1967).

2. U. Ingard, J. Acoust. Soc. Am. 44, 1155-1156 (1968).

3. A. G. Galaitsis, Ph. D. Thesis, Department of Physics, M. I. T., August 1972, seeSec. 2.1.

4. A. G. Galaitsis, Quarterly Progress Report No. 106, 1Research Laboratory of Elec-tronics, AI.I.T., July 15, 1972, pp. 44-47.

QPR No. 108